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When they said oxygen levels go down, they presumably meant that the partial pressure goes down. That’s what people probably think of anyway since that necessitates supplemental oxygen (if you are high enough).
The supplemental oxygen is needed because the total pressure is down, not because of gas separation. Humans need some number of moles of oxygen per minute, and at the top of Everest they just cannot ingest that number fast enough because the total pressure is so low. The percentage of oxygen in the total gas mixture is also lower on Everest than the 21% at sea level due to the aforementioned specific density separation, but that effect is much smaller and insignificant compared to the total pressure shortfall. This is why the grandparent comment objected to mentioning it - this is an interesting fact on its own, but irrelevant to the question of supplemental oxygen on Everest.
Yeah, on rereading I see that I originally missed the specific mention of densities causing them to separate (instead of just “less oxygen”), so I now agree with the objection.